The symbol \hookrightarrow usually denotes a split monomorphism. This generalizes the notion of an embedding. Using Mariano's notation, this means that it's a map that is split mono in the homotopy category, which means that it admits a (I always forget if it's left or right) homotopy inverse.
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Harry GindiJan 12 '10 at 3:41

5

The symbol denotes whatever the author tells you it will denote in his comments about notation, and there is a special place in hell for users of unexplained notation. I have never used the hooked arrow to mean anything but an inclusion map.
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Mariano Suárez-Alvarez♦Jan 12 '10 at 3:47

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Homotopy theorists are likely to interpret the hooked arrow with a tilde as "acyclic cofibration", which in general neither implies nor is implied by "inclusion which is a homotopy equivalence" (though it's certainly a similar notion; for example they agree for inclusions of a subcomplex of a CW complex).
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Reid BartonJan 12 '10 at 3:50

Dear Mariano, you write "The symbol denotes whatever the author tells you it will denote".This is practically Humpty Dumpty's reply to Alice in Through The Looking-Glass "When I use a word [...] it means just what I choose it to mean". I am sure it is your ever grinning Cheshire cat who whispered that in your ear.
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Georges ElencwajgJan 12 '10 at 8:29